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Properties

Label 340.2.bi.a.37.7
Level $340$
Weight $2$
Character 340.37
Analytic conductor $2.715$
Analytic rank $0$
Dimension $72$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [340,2,Mod(37,340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("340.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(340, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 4, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.bi (of order \(16\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.71491366872\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label 37.7
Character \(\chi\) \(=\) 340.37
Dual form 340.2.bi.a.193.7

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.25876 - 0.841077i) q^{3} +(-1.69071 - 1.46339i) q^{5} +(4.99604 - 0.993774i) q^{7} +(-0.270982 + 0.654209i) q^{9} +(-0.0182977 - 0.0919885i) q^{11} -1.69499 q^{13} +(-3.35902 - 0.420042i) q^{15} +(-0.808285 - 4.04310i) q^{17} +(-0.555392 - 1.34083i) q^{19} +(5.45298 - 5.45298i) q^{21} +(4.72214 - 7.06718i) q^{23} +(0.716983 + 4.94833i) q^{25} +(1.09518 + 5.50584i) q^{27} +(-1.26422 - 1.89204i) q^{29} +(-1.33938 + 6.73352i) q^{31} +(-0.100402 - 0.100402i) q^{33} +(-9.90112 - 5.63097i) q^{35} +(5.00912 + 7.49668i) q^{37} +(-2.13358 + 1.42561i) q^{39} +(0.409242 - 0.612474i) q^{41} +(-8.47887 + 3.51206i) q^{43} +(1.41551 - 0.709523i) q^{45} +9.58734i q^{47} +(17.5057 - 7.25109i) q^{49} +(-4.41800 - 4.40947i) q^{51} +(-3.54463 + 8.55750i) q^{53} +(-0.103679 + 0.182302i) q^{55} +(-1.82685 - 1.22066i) q^{57} +(-8.64350 - 3.58026i) q^{59} +(-1.11782 - 0.746906i) q^{61} +(-0.703702 + 3.53775i) q^{63} +(2.86573 + 2.48042i) q^{65} +(1.45408 - 1.45408i) q^{67} -12.8676i q^{69} +(7.11492 + 1.41525i) q^{71} +(0.826206 + 0.164343i) q^{73} +(5.06443 + 5.62572i) q^{75} +(-0.182832 - 0.441395i) q^{77} +(-5.67544 + 1.12892i) q^{79} +(4.50727 + 4.50727i) q^{81} +(-1.30417 - 0.540204i) q^{83} +(-4.55006 + 8.01854i) q^{85} +(-3.18271 - 1.31832i) q^{87} +(-2.30681 - 2.30681i) q^{89} +(-8.46822 + 1.68443i) q^{91} +(3.97745 + 9.60242i) q^{93} +(-1.02316 + 3.07971i) q^{95} +(-11.2332 - 2.23441i) q^{97} +(0.0651380 + 0.0129568i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 24 q^{15} + 8 q^{25} + 48 q^{27} - 32 q^{31} - 16 q^{33} - 32 q^{37} + 32 q^{39} - 40 q^{41} + 32 q^{53} - 16 q^{55} + 72 q^{57} - 112 q^{59} - 48 q^{63} + 32 q^{67} - 16 q^{71} - 96 q^{73} + 24 q^{75}+ \cdots - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/340\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(171\) \(241\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(e\left(\frac{1}{16}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.25876 − 0.841077i 0.726746 − 0.485596i −0.136334 − 0.990663i \(-0.543532\pi\)
0.863080 + 0.505067i \(0.168532\pi\)
\(4\) 0 0
\(5\) −1.69071 − 1.46339i −0.756107 − 0.654448i
\(6\) 0 0
\(7\) 4.99604 − 0.993774i 1.88833 − 0.375611i 0.891350 − 0.453316i \(-0.149759\pi\)
0.996976 + 0.0777048i \(0.0247592\pi\)
\(8\) 0 0
\(9\) −0.270982 + 0.654209i −0.0903274 + 0.218070i
\(10\) 0 0
\(11\) −0.0182977 − 0.0919885i −0.00551695 − 0.0277356i 0.977929 − 0.208939i \(-0.0670009\pi\)
−0.983446 + 0.181203i \(0.942001\pi\)
\(12\) 0 0
\(13\) −1.69499 −0.470105 −0.235052 − 0.971983i \(-0.575526\pi\)
−0.235052 + 0.971983i \(0.575526\pi\)
\(14\) 0 0
\(15\) −3.35902 − 0.420042i −0.867295 − 0.108454i
\(16\) 0 0
\(17\) −0.808285 − 4.04310i −0.196038 − 0.980596i
\(18\) 0 0
\(19\) −0.555392 − 1.34083i −0.127416 − 0.307609i 0.847279 − 0.531148i \(-0.178239\pi\)
−0.974695 + 0.223539i \(0.928239\pi\)
\(20\) 0 0
\(21\) 5.45298 − 5.45298i 1.18994 − 1.18994i
\(22\) 0 0
\(23\) 4.72214 − 7.06718i 0.984634 − 1.47361i 0.107010 − 0.994258i \(-0.465872\pi\)
0.877624 − 0.479350i \(-0.159128\pi\)
\(24\) 0 0
\(25\) 0.716983 + 4.94833i 0.143397 + 0.989665i
\(26\) 0 0
\(27\) 1.09518 + 5.50584i 0.210767 + 1.05960i
\(28\) 0 0
\(29\) −1.26422 − 1.89204i −0.234760 − 0.351343i 0.695320 − 0.718700i \(-0.255263\pi\)
−0.930080 + 0.367357i \(0.880263\pi\)
\(30\) 0 0
\(31\) −1.33938 + 6.73352i −0.240560 + 1.20938i 0.651919 + 0.758288i \(0.273964\pi\)
−0.892479 + 0.451088i \(0.851036\pi\)
\(32\) 0 0
\(33\) −0.100402 − 0.100402i −0.0174777 − 0.0174777i
\(34\) 0 0
\(35\) −9.90112 − 5.63097i −1.67360 − 0.951808i
\(36\) 0 0
\(37\) 5.00912 + 7.49668i 0.823495 + 1.23245i 0.969968 + 0.243232i \(0.0782077\pi\)
−0.146473 + 0.989215i \(0.546792\pi\)
\(38\) 0 0
\(39\) −2.13358 + 1.42561i −0.341647 + 0.228281i
\(40\) 0 0
\(41\) 0.409242 − 0.612474i 0.0639128 − 0.0956523i −0.798137 − 0.602477i \(-0.794181\pi\)
0.862049 + 0.506824i \(0.169181\pi\)
\(42\) 0 0
\(43\) −8.47887 + 3.51206i −1.29302 + 0.535585i −0.919883 − 0.392193i \(-0.871716\pi\)
−0.373133 + 0.927778i \(0.621716\pi\)
\(44\) 0 0
\(45\) 1.41551 − 0.709523i 0.211012 − 0.105769i
\(46\) 0 0
\(47\) 9.58734i 1.39846i 0.714898 + 0.699229i \(0.246473\pi\)
−0.714898 + 0.699229i \(0.753527\pi\)
\(48\) 0 0
\(49\) 17.5057 − 7.25109i 2.50081 − 1.03587i
\(50\) 0 0
\(51\) −4.41800 − 4.40947i −0.618644 − 0.617449i
\(52\) 0 0
\(53\) −3.54463 + 8.55750i −0.486893 + 1.17546i 0.469382 + 0.882995i \(0.344477\pi\)
−0.956275 + 0.292468i \(0.905523\pi\)
\(54\) 0 0
\(55\) −0.103679 + 0.182302i −0.0139801 + 0.0245816i
\(56\) 0 0
\(57\) −1.82685 − 1.22066i −0.241972 − 0.161681i
\(58\) 0 0
\(59\) −8.64350 − 3.58026i −1.12529 − 0.466110i −0.259111 − 0.965847i \(-0.583430\pi\)
−0.866177 + 0.499738i \(0.833430\pi\)
\(60\) 0 0
\(61\) −1.11782 − 0.746906i −0.143123 − 0.0956316i 0.481947 − 0.876200i \(-0.339930\pi\)
−0.625070 + 0.780569i \(0.714930\pi\)
\(62\) 0 0
\(63\) −0.703702 + 3.53775i −0.0886581 + 0.445714i
\(64\) 0 0
\(65\) 2.86573 + 2.48042i 0.355449 + 0.307659i
\(66\) 0 0
\(67\) 1.45408 − 1.45408i 0.177644 − 0.177644i −0.612684 − 0.790328i \(-0.709910\pi\)
0.790328 + 0.612684i \(0.209910\pi\)
\(68\) 0 0
\(69\) − 12.8676i − 1.54907i
\(70\) 0 0
\(71\) 7.11492 + 1.41525i 0.844385 + 0.167959i 0.598283 − 0.801285i \(-0.295850\pi\)
0.246103 + 0.969244i \(0.420850\pi\)
\(72\) 0 0
\(73\) 0.826206 + 0.164343i 0.0967001 + 0.0192348i 0.243203 − 0.969975i \(-0.421802\pi\)
−0.146503 + 0.989210i \(0.546802\pi\)
\(74\) 0 0
\(75\) 5.06443 + 5.62572i 0.584791 + 0.649602i
\(76\) 0 0
\(77\) −0.182832 − 0.441395i −0.0208356 − 0.0503016i
\(78\) 0 0
\(79\) −5.67544 + 1.12892i −0.638537 + 0.127013i −0.503733 − 0.863859i \(-0.668040\pi\)
−0.134804 + 0.990872i \(0.543040\pi\)
\(80\) 0 0
\(81\) 4.50727 + 4.50727i 0.500808 + 0.500808i
\(82\) 0 0
\(83\) −1.30417 − 0.540204i −0.143151 − 0.0592951i 0.309958 − 0.950750i \(-0.399685\pi\)
−0.453109 + 0.891455i \(0.649685\pi\)
\(84\) 0 0
\(85\) −4.55006 + 8.01854i −0.493523 + 0.869733i
\(86\) 0 0
\(87\) −3.18271 − 1.31832i −0.341222 − 0.141339i
\(88\) 0 0
\(89\) −2.30681 − 2.30681i −0.244522 − 0.244522i 0.574196 − 0.818718i \(-0.305315\pi\)
−0.818718 + 0.574196i \(0.805315\pi\)
\(90\) 0 0
\(91\) −8.46822 + 1.68443i −0.887711 + 0.176577i
\(92\) 0 0
\(93\) 3.97745 + 9.60242i 0.412443 + 0.995724i
\(94\) 0 0
\(95\) −1.02316 + 3.07971i −0.104974 + 0.315972i
\(96\) 0 0
\(97\) −11.2332 − 2.23441i −1.14055 − 0.226870i −0.411555 − 0.911385i \(-0.635014\pi\)
−0.728999 + 0.684515i \(0.760014\pi\)
\(98\) 0 0
\(99\) 0.0651380 + 0.0129568i 0.00654662 + 0.00130220i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 340.2.bi.a.37.7 yes 72
5.3 odd 4 340.2.bd.a.173.7 yes 72
17.6 odd 16 340.2.bd.a.57.7 ✓ 72
85.23 even 16 inner 340.2.bi.a.193.7 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.bd.a.57.7 ✓ 72 17.6 odd 16
340.2.bd.a.173.7 yes 72 5.3 odd 4
340.2.bi.a.37.7 yes 72 1.1 even 1 trivial
340.2.bi.a.193.7 yes 72 85.23 even 16 inner